Electrical – Can Panel Handle Hot Tub – 50 amps

1. Number 6 AWG can be protected at 70 amps if the equipment at both ends is rated for 75°C yes to your wire colors.
2. Yes
3. If the new sub is attached to the main or right next to it, it does not need a ground rod. The NEC requires a grounding electrode at a separate building or structure containing more than one branch circuit. The single circuit to the hot tub does not require a ground rod. The equipment goundung conductor with the circuit is sufficient.

Nipples between panels of 24" or less do not require you to derate the wire ampacity.

When attaching a sub-panel I would not limit myself to 1" conduit. I would use a 1 1\2" or 2" nipple to allow more wire to pass between the panels. Just a personal preference though.

Good luck and stay safe!

The clicking noise sounds like a contactor or power relay. On the tubs I have worked on there has been a safety pressure switch on the pump, no pressure the switch opens and the heat contactor drops out. Other possible problems include a rusty contactor trying to pull in, loud buzz because it's not fully closing. I would check the pressure switch and look for a possible thermal overtemp switch also. Loose or corroded connections are quite possible also. So an inspection of the connections could also identify the problem. The contactor issue needs to be fixed but you can test the heater with the power off, measure the terminals with an ohm meter the value will probably be 10-30 ohms if it shows open it is bad. The resistance is based on the size of the heat element so it could be lower or higher but will probably be close to 10-30 ohms. I have bypassed both pressure switches and over temp switches to verify that is the problem but once solved never leave a safety bypassed.